How do I apply a texture in Blender Render without it being influenced by the object it's applied to?
Its quite easy, just pick the "Window" coordinate in the texture panel under "Mapping"



One bit of advice though is that during animation, when you move the object or camera, the texture will move with it. i.e. your "pixels" wont stay in the same spot, so you will potentially have a flicker in your "dithering".

For stills though this is totally fine.